Tseng's Algorithm with Extrapolation from the Past Endowed with Variable Metrics and Error Terms

TL;DR

This paper introduces a variable metric, extrapolated version of Tseng's forward-backward-forward algorithm with error terms, applicable to monotone inclusion problems and demonstrated in image deblurring.

Contribution

It extends Tseng's algorithm by incorporating variable metrics, extrapolation, and error terms, and applies it to primal-dual problems in Hilbert spaces.

Findings

Converges for monotone inclusion problems with variable metrics.

Effective in image deblurring applications.

Generalizes existing primal-dual algorithms.

Abstract

In this paper, we propose a variable metric version of Tseng's algorithm (the forward-backward-forward algorithm: FBF) combined with extrapolation from the past that includes error terms for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. This can be seen as the optimistic gradient descent ascent (OGDA) algorithm endowed with variable metrics and error terms. Primal-dual algorithms are also proposed for monotone inclusion problems involving compositions with linear operators. The primal-dual problem occurring in image deblurring demonstrates an application of our theoretical results.